As the title says - blue dwarf stars don't exist. At the moment, they are only theoretical, as the time necessary for their formation is longer than the current age of the Universe.
The possibility of their eventual existence is predicted by theoretical models as being a stage of development of red dwarves. Generally, when stars age, their brightness increases. A mechanism has to come into play then, to maintain the star's equilibrium. Most stars do this by increasing in size, becoming red giants.
Studying the evolution of red dwarf stars, it is thought that, for ones that are one-quarter the mass of our Sun, or less, the method that would be employed is to increase the surface temperature instead, whilst staying roughly the same size.
As the general rule is, the hotter the star, the bluer it is, these red dwarves will therefore become blue.
The reason that no blue dwarf stars exist yet is due to the length of time that red dwarves remain as main-sequence stars. They tend to fuse their hydrogen supply very slowly, thus having lives that can last for trillions of years. As the estimated age of the Universe is 13.8 billion years, it is simply not old enough for any red dwarves to have exhausted almost all their hydrogen, heralding their transformation into blue dwarves.
Extrapolating the theory even further, it is thought that, once a blue dwarf's hydrogen supply is completely used up, it will become a white dwarf, eventually degenerating into a black dwarf.